Research Article Optimal Multiuser MIMO Linear Precoding with LMMSE Receiver

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1 Hindawi Publishing Corporation EURASIP Journal on Wireless Communications and Networing Volume 009, Article ID 9768, 0 pages doi:055/009/9768 Research Article Optimal Multiuser MIMO Linear Precoding with LMMSE Receiver Fang Shu, Wu Gang, and Li Shao-Qian National Key Lab of Communication, University of Electronic Science and Technology of China, Chengdu 673, China Correspondence should be addressed to Fang Shu, fangshu@uestceducn Received January 009; Revised May 009; Accepted 9 June 009 Recommended by Cornelius van Rensburg The adoption of multiple antennas both at the transmitter and the receiver will explore additional spatial resources to provide substantial gain in system throughput with the spatial division multiple access SDMA) technique Optimal multiuser MIMO linear precoding is considered as a ey issue in the area of multiuser MIMO research The challenge in such multiuser system is designing the precoding vector to imize the system capacity An optimal multiuser MIMO linear precoding scheme with LMMSE detection based on particle swarm optimization is proposed in this paper The proposed scheme aims to imize the system capacity of multiuser MIMO system with linear precoding and linear detection This paper explores a simplified function to solve the optimal problem With the adoption of particle swarm optimization algorithm, the optimal linear precoding vector could be easily searched according to the simplified function The proposed scheme provides significant performance improvement comparing to the multiuser MIMO linear precoding scheme based on channel bloc diagonalization method Copyright 009 Fang Shu et al This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original wor is properly cited Introduction In recent years, with the increasing demand of transmitting high data rates, the Multiple-Input Multiple-Output) MIMO technique, a potential method to achieve high capacity has attracted enormous interest [, ] When multiple antennas are equipped at both base stations BSs) and mobile stations MSs), the space dimension can be exploited for scheduling multi-user transmission besides time and frequency dimension Therefore, the traditional MIMO technique focused on point-to-point single-user MIMO SU-MIMO) has been extended to the point-tomultipoint multi-user MIMO MU-MIMO) technique [3, 4] It has been shown that time division multiple access TDMA) systems can not achieve sum rate capacity of MU- MIMO system of broadcast channel BC) [5] while MU- MIMO with spatial division multiple access SDMA) could, where one BS communicates with several MSs within the same time slot and the same frequency band [6, 7] MU- MIMO based on SDMA improves system capacity taing advantage of multi-user diversity and precanceling of multiuser interference at the transmitter Traditional MIMO technique focuses on point-to-point transmission as the STBC technique based on space-time coding and the VBLAST technique based on spatial multiplexing The former one can efficiently combat channel fading while its spectral efficiency is low [8, 9] The latter one could transmit parallel data streams, but its performance will be degraded under spatial correlated channel [0, ] When the MU-MIMO technique is adopted, both the multiuser diversity gain to improve the BER performance and the spatial multiplexing gain to increase the system capacity will be obtained Since the receive antennas are distributed among several users, the spatial correlation will effect less on multi-user MIMO system Besides, because the multiuser MIMO technique utilizes precoding at the transmit side to precanceling the cochannel interference CCI), so the complexity of the receiver can be significantly simplified However multi-user CCI becomes one of the main obstacles to improve MU-MIMO performance The challenge is that the receiving antennas that are associated with different users are typically unable to coordinate with each other By mitigating or ideally completely eliminating CCI, the BS exploits the channel state information CSI) available at the

2 EURASIP Journal on Wireless Communications and Networing transmitter to cancel the CCI at the transmitter It is essential to have CSI at the BS since it allows joint processing of all users signals which results in a significant performance improvement and increased data rates The sum capacity in a multiuser MIMO broadcast channel is defined as the imum aggregation of all the users data rates For Gaussion MIMO broadcast channels BCs), it was proven in [] that Dirty Paper Coding DPC) can achieve the capacity region The optimal precoding of multi-user MIMO is based on dirty paper coding DPC) theory with the nonlinear precoding method DPC theory proves that when a transmitter has advance nowledge of the interference, it could design a code to compensate for it It is developed by Costa which can eliminate the interference by iterative precoding at the transmitter and achieve the broadcast MIMO channel capacity [3, 4] The famous Tomlinson-Harashima precoding THP) is the non-linear precoding based on DPC theory It is first developed by Tomlinson [5] and Miyaawa and Harashima [6] independently and then has become the Tomlinson- Harashima precoding THP) [7 0] to combat the multiuser cochannel interference CCI) with non-linear precoding Although THP performs well in a multi-user MIMO scenario, deploying it in real-time systems is difficult because of its high complexity of the precoding at the transmitter Many suboptimal MU-MIMO linear precoding techniques have emerged recently, such as the channel inversion method [] and the bloc diagonalization BD) method [ 4] Channel inversion method [5] employs some traditional MIMO detection criterions, such as the Zero Forcing ZF) and Minimum Mean Squared Error MMSE), precoding at the transmitter to suppress the CCI Channel inversion method based on ZF can suppress CCI completely; however it may lead to noise amplification since the precoding vectors are not normalized Channel inversion method based on MMSE compromises the noise and the CCI, and outperforms ZF algorithm, but it still cannot obtain good performance BD method decomposes a multi-user MIMO channel into multiple single user MIMO channels in parallel to completely cancel the CCI by maing use of the null space With BD, each users precoding matrix lies in the null space of all other users channels, and the CCI could be completely canceled The generated null space vectors are normalized vectors, which could avoid the noise amplification problem efficiently So BD method performs much better than channel inversion method However, since BD method just aims to cancel the CCI and suppress the noise, its precoding gain is not optimized It is obvious that the CCI, the noise, and the precoding gain are the factors affecting on the performance of the preprocessing MU-MIMO The above linear precoding methods just tae one factor into account without entirely consideration A rate imization linear precoding method is proposed in [6] This method aims to imize the sum rate of the MU-MIMO system with linear preprocessing However,theoptimizedfunctionin[6] is too complex to compute In this paper, we solve the optimal linear precoding with linear MMSE receiver problem in a more simplified way MS4 H 4 H 3 MS3 BS H H MS MS Figure : The configuration of MU-MIMO system An optimal MU-MIMO linear precoding scheme with linear MMSE receiver based on particle swarm optimization PSO) is proposed in this paper PSO algorithm has been used in many complex optimization tass, especially in solving the optimization of continuous space [7, 8] In this paper, PSO is firstly introduced into MIMO research to solve some optimization issues The adoption of PSO to MIMO system provides a new method to solve the MIMO processing problem In this paper, we first analyze the optimal linear precoding vector with linear MMSE receiver and establish a simplified function to measure the optimal linear precoding problem Then, we employ the novel PSO algorithm to search the optimal linear precoding vector according to the simplified function The proposed scheme obtains significant MU-MIMO system capacity and outperforms the channel bloc diagonalization method This paper is organized into seven parts The system model of MU-MIMO is given in Section Then the analysis of optimal linear precoding with linear MMSE receiver is given in Section 3 The particle swarm optimization algorithm is given in Section 4 InSection 5, the proposed optimal linear precoding MU-MIMO scheme with LMMSE detection based on particle swarm optimization is introduced In Section 6, the simulation results and comparisons are given Conclusions are drawn in the last section The channel bloc diagonalization algorithm is given in the appendix System Model of MU-MIMO The MU-MIMO system could transmit data streams of multiple users of the same cellular at the same time and the same frequency resources as Figure shows We consider an MU-MIMO system with one BS and K MS, where the BS is equipped with M antennas and each MS with N antennas, as shown in Figure The point-to multipoint MU-MIMO system is employed in downlin transmission Because MU-MIMO aims to transmit data streams of multiple-users at the same time and frequency resources, we discuss the algorithm at single-carrier, for each subcarrier of the multicarrier system, and it is processed as same as the single-carrier case Since OFDM technique deals the frequency selective fading as flat fading, we model the channel as the flat fading MIMO channel: h, h,m H = ) h N, h N,M

3 EURASIP Journal on Wireless Communications and Networing 3 S S S K T T T K M H H K Y N N Y Y K N Figure : The bloc diagram of MU-MIMO system G G G s s s In order to simplify the analysis, the power allocation is assumed as equal β = p /N 0 = p 0 /KN 0, and linear MMSE MIMO detection is used in this paper as G = h H H H H + βi ), N 6) where ) indicates the inverse of the matrix, ) H denotes the matrix conjugation transposition, and I N is the N N identity matrix: [ ] h = H T = H = [H W], 7) where H is the MIMO channel matrix of user h i,j indicates the channel impulse response coupling the jth transmit antenna to the ith receive antenna Its amplitude obeys independent and identically Rayleigh-distribution Data streams of K K M) users are precoded by their precoding vectors T = K) before transmission T is the M normalized precoding vector for user with T H T = The received signal at the th user is y = H T p s + H K i=,i T i pi s i + n p = p 0 = where y is the received signal of user The elements of additive noise n obey distribution CN0, N 0 ) that are spatially and temporarily white p is the transmit signal power of the th data stream, and p 0 is the total transmit power The received signal at the th user can also be expressed as y = H Ws + n W = [T T T K ] 3) [ p ] T s = s p s pk s K where s is the transmitted symbol vector with K data streams, W is the precoding matrix with K precoding vectors, and [ ] T denotes the matrix transposition: H = H W 4) Thechannelmatrix H can be assumed as the virtual channel matrix of user after precoding At the receiver, a linear receiver G is exploited to detect the transmit signal for the user The detected signal of the th user is ŝ = G y 5) The linear receiver G can be designed by ZF or MMSE criteria, and linear MMSE will obtain better performance where [ ] denotes the th column of the matrix Then the detected SINR for the user with the linear detection is β G H T SINR = Ki=, i β G H T i + G β G H T / G = Ki=,, i β G H T i / G + where denotes the matrix two-norm Because the nonnormalized precoding vector will amplify the noise at the receiver, the precoding vectors T are assumed to be normalized as follow: for =,, K 8) T = 9) 3 Optimal Multiuser MIMO Linear Precoding We assume that the MIMO channel matrices H =,, K) are available at the BS It can be achieved either by channel reciprocity characteristics in time-division-duplex TDD) mode or by feedbac in frequency-division-duplex FDD) mode And the channel matrix H is nown at the receiver through channel estimation We just discuss the equal power allocation case in this paper The optimal power allocation is achieved through water-filling according to the SINRofeachuser The MIMO channel of user can be decomposed by the singular value decomposition SVD) as H = U Σ V H 0) If we apply T = [V ] to precode for user, it obtains the imal precoding gain as follow Lemma One has H T = H [V ] =, ) where [V ] denotes the first column of V, and denotes the imal singular value of H

4 4 EURASIP Journal on Wireless Communications and Networing Proof One has H T = H T ) H H T ) = H [V ] ) H H [V ] ) ) Because we assume that [V ] H [V i ] = 0 i ), and [V ] is the unit vector with [V ] H [V ] =, so W = [[V ],[V ],,[V K ] ] is an unitary matrix with WW H = I M, ) So = ) H ) [U ] [U ] G = H T ) H H H H + β I M ) H = [U ] U Σ UH + ) β I M H T =, 3) where [U ] denotes the first column of unitary matrix U = [U ] H U Σ + β I M ) U H ) ) Thus, precoding with the singular vector corresponding to the imal singular value is an initial thought to obtain good performance However, if the singular vector is directly used at the transmitter as the precoding vector, the CCI will be large, and the performance will be degraded severely Only for the special case that the MIMO channel among all these users are orthogonal that the CCI will be zero if we directly use the singular vector of each user as its precoding vector But in realistic case, the transmit users channels are always nonorthogonal, and so the singular vector could not be utilized directly We have drawn some analysis as follow )Ideal channel case The ideal channel case is that the MIMO channels of transmitting users are orthogonal There is [V ] ) H [V i ] = 0 i ) 4) If we apply T = [V ] to precode for user, the imal precoding gain will be obtained as 3) shows, and the CCI will turn to zero as follow Lemma One has Proof One has i=,i Ki=, i β G H T i G G H T i = K i=,i = 0 5) G H [V i ], 6) G = H T ) H H H H + β I M), 7) H T = H [V ] = [U ], 8) G = [U ] ) H H W)H W) H + β I M), 9) H W)H W) H = U Σ V H WWH V Σ U H 0) Since [U ] ) H U H ) = [, 0,,0],so ) G = [U ] H U H Σ + ) β I M U = = ) +/β [ U H ] ) +/β [U ] H, where [U H ] denotes the first row of U H Also i=,i G H T i = i=,i ) +/β [U ] H H T i = i=,i ) +/β [U ] H U Σ V H [V i] ) 3) Since [V ] ) H [V i ] =0, so V H [V i] = [0, a,, a K ] T Combining [U ] ) H U = [, 0,, 0], there is so i=,i G H T i = K i=,i Ki=, i β G H T i G G H [V i ] = 0 4) = 0 5) After linear MMSE detection at the receiver, user obtains the imal SINR as follows Lemma 3 One has SINR β G H T = G = β ) 6)

5 EURASIP Journal on Wireless Communications and Networing 5 Proof One has SINR β G H T ) H G H T = G G H According to 3)and) ) H ) G H T G H T ) 7) Since we assume that [V ] H [V i ] = i ), so W = [[V ],[V ],,[V K ] ]is W = 33) = = G G H = H ) [U ] H [U ] +/β ) [U ] H [U ] +/β ) ) +/β SINR = β ) ) +/β 8) )Ill channel case The ill channel case is that all these transmitting users channels are highly correlated There is [V ] ) H [V i ] = i ) 9) If we still apply T = [V ] to precode for user, the multiuser CCI will be very large, and the system performance will be degraded severely The SINR after MMSE detection with equal power allocation for user is as follows Lemma 4 One has β ) SINR = Ki=,i β ) + 30) Proof Since we have proven that when T = [V ] to precode for user, then β G H T / G = β ),so β G H T / G SINR = Ki=,i β G H T i / G + β ) 3) = Ki=,i β G H T i / G + According to 9) ) H G = [U ] U Σ V H WWH V Σ U H + β I M) 3) Let the diagonal matrix Σ = Σ V H WWH V Σ, and so there is ) H G = [U ] U Σ + ) ) β I M U H 34) Since [U ] ) H U H ) = [, 0,,0],so G = [ ] U H +/β = +/β [U ] H, 35) where indicates the first diagonal element of the diagonal matrix Σ So there is G H T i = +/β [U ] H H T i = +/β [U ] H U Σ V H [V i] 36) Since [V ] ) H [V i ] =, so V H [V i] = [, a,, a K ] T Combining [U ] ) H U = [, 0,, 0], there is G H T i = G = G G H ), +/β ) =, +/β Ki=,i β G H T = β ) G i=,i So the SINR for user is SINR = β ) Ki=,i β 37) ) + 38) 3) Practical case The practical case is that the transmitting users channels are neither orthogonal nor ill There is [V ] ) H [V i ] 0 i ) [V ] ) H [V i ] i ) 39) The practical case is usually in realistic environment If we apply T T [V ] ) to precode for user, then

6 6 EURASIP Journal on Wireless Communications and Networing CDF Capacity bps/hz) Channel inversion with ZF precoder, 4R 4 data streams Channel inversion with MMSE, precoder, 4TR 4 data streams Proposed MU-MIMO 4TR4 data streams Figure 3: The system capacity CDF comparison of the two schemes ξ = T H [V ] can be the parameter to measure the precoding gain, and ρ i = T H [V i] can be the parameter to measure the CCI The SINR for user according to the above analysis can be approximated denoted as SINR β ξ ) Ki,i= β ρ i ) + T H [V ] ) = Ki,i= T H [V i] ) +/β 40) The system capacity is related to SINR of the transmit users, =,, K) So in order to obtain the system capacity, we should obtain the SINR Thus, when the optimal precoding vector is obtained by the PSO algorithm, the system capacity could be calculated by 4) The system capacity of the MU-MIMO system can be indicated as C MU = log + SINR ) 4) = We aim to imize the system capacity of the MU- MIMO system in this paper The optimal MU-MIMO linear precoding vector for the MU-MIMO system is the vector that can imize the SINR at each receiver as T = arg T U T H log + [V ] ) Ki =,i= T H [V i] ) +/β 4) where U denotes the unitary vector that U H U = I From the above equation, it is clear that if we want to imize the system capacity of MU-MIMO, then the SINR of each user should be imized The SINR of user is associated with three parameters as the singular vector correspond to the imal singular value of all users and the noise 4 The Particle Swarm Optimization Algorithm Particle swarm optimization algorithm was originally proposed by Kennedy and Eberhart[7] in 995 It searches the optimal problem solution through cooperation and competition among the individuals of population Imagine a swarm of bees in a field Their goal is to find in the field the location with the highest density of flowers Without any prior nowledge of the field, the bees begin in random locations with random velocities looing for flowers Each bee can remember the location that is found the most flowers and somehow nows the locations where the other bees found an abundance of flowers Torn between returning to the location where it had personally found the most flowers, or exploring the location reported by others to have the most flowers, the ambivalent bee accelerates in both directions to fly somewhere between the two points There is a function or method to evaluate the goodness of a position as the fitness function Along the way, a bee might find a place with a higher concentration of flowers than it had found previously Constantly, they are checing the concentration of flowers and hoping to find out the absolute highest concentration of flowers Suppose that the size of swarm and the dimension of search space are C and D,respectively Each individual in the swarm is referred to as a particle The location and velocity of particle i i =,, C) are represented as the vector x i = [x i, x i,, x id ] T and v i = [v i, v i,, v id ] T Eachbee remembers the location where it personally encountered the most flowers which is denoted as P i = [p i, p i,, p id ] T, which is the flight experience of the particle itself The highest concentration of flowers discovered by the entire swarm is denoted as P g = [p g, p g,, p gd ] T, which is the flight experience of all particles Each particle is searching for the best location according to P i and P g The particle i updates its location and velocity according to the following two formulas [7]: v t+ id = wv t id + c ϕ p t id x t id) + c ϕ p t gd x t id x t+ id = x t id + vt+ id ) 43) where t is the current iteration number; vid t and xt id + denote the velocity and location of the particle i in the dth dimensional direction pid t is the individual best location of particle i in the dth dimensional direction, pgd t is the population best location in the dth dimensional direction ϕ and ϕ are the random numbers between 0 and, c and c are the learning factors, and w is the inertia factor Learning factors determine the relative pull of P t i and P t g that usually content c = c = Inertia factor determines to what extent the particle remains along its original course unaffected by the pull of P t g and P t i that is usually between 0 and After this process is carried out for each particle in the swarm, the

7 EURASIP Journal on Wireless Communications and Networing 7 process is repeated until reaching the imal iteration or the termination criteria are met 5 The Optimal Linear Precoding Multiuser MIMO with LMMSE Detection Based on Particle Swarm Optimization With the adoption of PSO algorithm and the simplified function 40), the optimal linear precoding vector T =,, K) could be easily searched The proposed optimal MU-MIMO linear precoding scheme based on PSO algorithm will search the optimal precoding vector for each user following 6 steps ) The BS obtains,[v ] and β of each user ) The BS employs the PSO algorithm to search the optimal linear precoding vector for each user For user, the PSO algorithm sets the imal iteration number I and a group of M dimensional particles with the initial velocity vi, = [v i,, v i,,, v im, ]T and the initial location xi, = [x i,, x i,,, x im, ]T for particle i i =,, C) In order to accelerate the searching process, the initial location xi, could be initialized as [V ], while the initial velocity vi, could be produced randomly The real and imaginary parts of the initial velocity obey a normal distribution with mean zero and standard deviation one 3) The BS begins to search with the initial location xi, and velocity vi, The goodness of the location is measured by the following equation: fi, t = [xi, t )H [V ] ] Kj=,j [xi, t )H [V j ] ], 44) +/β where the fitness function fi, t indicates the obtained SINR for user precoded by xi, t The PSO algorithm finds P t i, and Pt g, that are individual best location and population best location measured by 44) for the next iteration P t i, denotes the individual best location which means the best location of particle i at the tth iteration of the th user P t g, denotes the population best location which means the best location of all particles at the tth iteration of the th user 4) For the tth iteration, the algorithm finds a P t i, and a P t g, The location and velocity for each particle will be updated according to 43) for the next iteration In order to obtain the normalized optimal precoding vector to suppress the noise, the location xi, t should be normalized in each iteration 5) When reaching the imal iteration number I, the algorithm stops, and P I g, is the obtained optimal precoding vector for user 6) For an MU-MIMO system with K users, the scheme will search the precoding vectors according to the above criteria for each user CDF Capacity bps/hz) Coordinate Tx-Rx BD 4TR 4 users Proposed MU-MI MO 4TR 4 users Figure 4: The system capacity CDF comparison of the two schemes 6 Simulation Results We simulated the proposed MU-MIMO scheme, the BD algorithm in [] Coordinate Tx-Rx BD), and the channel inversion algorithm in [5] in this paper to compare their performance under the same simulation environment Figure 3 is the system capacity comparison of the cumulative distribution function CDF) of the channel inversion algorithm with ZF precoder and MMSE precoder and the proposed MU-MIMO algorithm when M = 4, N =, p 0 /N 0 = 5 db with equation power allocation and MMSE detection at the receiver For channel inversion method, the BS transmits 4 date streams and users simultaneously with date stream for each user For the proposed MU-MIMO, the BS transmit 4 data streams and 4 users simultaneously with data stream for each user Figure 4 is the system capacity comparison of the CDF of the coordinated Tx-Rx BD algorithm and the proposed MU- MIMO algorithm when M = 4, K = 4, N =, p 0 /N 0 = 5 db with equation power allocation and MMSE detection at the receiver Figure 5 is the system capacity comparison of the CDF of the coordinated Tx-Rx BD algorithm and the proposed MU- MIMO algorithm when M = 4, K = 4, N = 4, p 0 /N 0 = 5 db with equation power allocation and MMSE detection at the receiver Both the simulation results of the proposed MU- MIMO scheme with PSO algorithm from Figure 3 to Figure 5 are based on the PSO parameters with the particle number C = 0 and the iteration number I = 0 It could be seen that the proposed MU-MIMO scheme can effectively increase the system capacity compared to the BD algorithm and channel inversion algorithm Figure 6 is the average BER performance of the proposed MU-MIMO scheme and the coordinated Tx-Rx BD algorithm with M = 4, K = 4, N = 4 Figure 7 is the average BER performance of the proposed MU-MIMO scheme and the coordinated Tx-Rx BD algorithm with M = 4, K = 4, N =

8 8 EURASIP Journal on Wireless Communications and Networing CDF Capacity bps/hz) Coordinate Tx-RxBD 4TR 4 users Proposed MU-MIMO 4TR 4 users BER SNR db) Coordinate Tx-RxBD 4TR 4 users Proposed MU-MIMO 4TR 4 users Figure 5: The system capacity CDF comparison of the two schemes Figure 7: The BER comparison of the two schemes BER BER SNR db) Coordinate Tx-RxBD 4TR 4 users Proposed MU-MIMO 4TR 4 users Figure 6: The BER comparison of the two schemes SNR db) Proposed MU-MIMO 4T4R, C = 0, I = 30 Proposed MU-MIMO 4T4R, C = 0, I = 0 Proposed MU-MIMO 4T4R, C = 0, I = 0 Proposed MU-MIMO 4T4R, C = 0, I = 5 Figure 8: The BER comparison of the two schemes with different C and I Both the schemes adopt equal power allocation, MMSE detection, QPSK, and no channel coding The proposed MU- MIMO scheme, with PSO algorithm from Figures 6 and 7 are based on the PSO parameters with the particle number C = 0 and the iteration number I = 0 From the simulation results, it is clear that the proposed MU-MIMO linear precoding with LMMSE detection based on particle swarm optimization scheme outperforms the BD algorithm and the channel inversion algorithm The reason lies in that the BD algorithm just aims to utilize the normalized precoding vector to cancel the CCI and suppress the noise The channel inversion algorithm also aims to suppress CCI and noise So the users transmit signal covariance matrices of these schemes are generally not optimal that are caused by the inferior precoding gain The proposed MU-MIMO optimal linear precoding scheme aims to find the optimal precoding vector to imize each users SINR at each receiver to improve the total system capacity Figure 8 shows the BER performance of the proposed MU-MIMO scheme with the same particle size and different iteration size when M = 4, K = 4, N = 4 It adopts equal power allocation, MMSE detection, QPSK, and no channel coding The particle number C is 0, and the iteration number scales from 5 to 30 We could see that when the iteration number is small, the proposed scheme could not obtain the best performance With the increase of the iteration number, more performance as well as the computational complexity will increase too However, when the iteration number is

9 EURASIP Journal on Wireless Communications and Networing 9 larger than 0 for this case, the algorithm could not obtain more performance gain Generally, for different case, the best iteration number is different The iteration number is related to the transmit antenna number M at the BS and the transmit user number K K M) With the increasing of M or K, the iteration number should increase in order to obtain the best performance 7 Conclusion This paper solves the optimal linear precoding problem with LMMSE detection for MU-MIMO system in downlin transmission A simplified optimal function is proposed and proved to imize the system capacity With the adoption of the particle swarm optimization algorithm, the optimal linear precoding vector with LMMSE detection for each user could be searched The proposed scheme can obtain significant system capacity improvement compared to the multi-user MIMO scheme based on channel bloc digonolization under the same simulation environment Appendix Coordinated Tx-Rx BD Algorithm Coordinated Tx-Rx BD algorithm is the improved BD algorithm It could solve the antenna constraint problem in traditional BD algorithm and extends the BD algorithm to arbitrary antenna configuration For a coordinated Tx- Rx BD algorithm with M transmit antennas at the BS, N receive antennas at the MS, and K users to be transmitted simultaneously, the algorithm follows 6 steps ) For j =,, K, compute the SVD H j = U j Σ j V H j A) ) Determine m j, which is the number of subchannels for each user In order to compare the two schemes fairly, m j = foreachuser 3) For j =,, K,letA j be the first m j columns of U j Calculate H j = A H j H j [ T H j = H H T j H T j+ H T ] T K A) 4) For j =,, K, computeṽ 0) j, the right null space of H j as H j = Ũ j Σ j [Ṽ) j Ṽ 0) ] H, j A3) where Ṽ ) j holds the first L j right singular vectors, Ṽ 0) j holds the last N L j right singular vectors and L j = ranh j ) 5) Compute the SVD H j Ṽ 0) j = U j Σ j 0 [ V ) j V 0) ] H j A4) 0 0 6) The precoding matrix W for the transmit users with average power allocation is [Ṽ0) W = V ) Acnowledgments Ṽ 0) V ) Ṽ 0) K V ) ] K A5) The project was supported by the National Natural Science Foundation of China ) and the Key Laboratory of Universal Wireless Communications Lab Beijing University of Posts and Telecommunications), Ministry of Education, China References [] G J Foschini and M J Gans, On limits of wireless communications in a fading environment when using multiple antennas, Wireless Personal Communications,vol6,no3,pp 3 335, 998 [] A Goldsmith, S A Jafar, N Jindal, and S Vishwanath, Capacity limits of MIMO channels, IEEE Journal on Selected Areas in Communications, vol, no 5, pp , 003 [3] QHSpencer,CBPeel,ALSwindlehurst,andMHaardt, An introduction to the multi-user MIMO downlin, IEEE Communications Magazine, vol 4, no 0, pp 60 67, 004 [4] C Windpassinger, R F H Fischer, T Vencel, and J B Huber, Precoding in multiantenna and multiuser communications, IEEE Transactions on Wireless Communications, vol 3, no 4, pp , 004 [5] N Jindal and A Goldsmith, Dirty-paper coding versus TDMA for MIMO broadcast channels, IEEE Transactions on Information Theory, vol 5, no 5, pp , 005 [6]PViswanathandDNCTse, Sumcapacityofthevector Gaussian broadcast channel and uplin-downlin duality, IEEE Transactions on Information Theory, vol49,no8,pp 9 9, 003 [7] S Vishwanath, N Jindal, and A Goldsmith, Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels, IEEE Transactions on Information Theory, vol 49, no 0, pp , 003 [8] E A Jorswiec and A Sezgin, Impact of spatial correlation on the performance of orthogonal space-time bloc codes, IEEE Communications Letters, vol 8, no, pp 3, 004 [9] S N Diggavi, On achievable performance of spatial diversity fading channels, IEEE Transactions on Information Theory, vol 47, no, pp , 00 [0] P Tarasa, H Minn, and V K Bhargava, Linear prediction receiver for differential space-time bloc codes with spatial correlation, IEEE Communications Letters, vol 7, no, pp , 003 [] H Bölcsei, D Gesbert, and A J Paulraj, On the capacity of OFDM-based spatial multiplexing systems, IEEE Transactions on Communications, vol 50, no, pp 5 34, 00 [] M H M Costa, Writing on dirty paper, IEEE Transactions on Information Theory, vol 9, no 3, pp , 983

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